Let $R$ be a compact, connected, orientable surface of genus $g$ with $n$ boundary components with $g \geq 2$, $n \geq 0$. Let $\mathcal{ N}(R)$ be the nonseparating curve graph, $\mathcal{ C}(R)$ be the curve graph and $\mathcal{ HT}(R)$ be the Hatcher-Thurston graph of $R$. We prove that if $\lambda : \mathcal{ N}(R) \rightarrow\mathcal{ N}(R)$ is an edge-preserving map, then $\lambda$ is induced by a homeomorphism of $R$. We prove that if $\theta : \mathcal{ C}(R) \rightarrow \mathcal{ C}(R)$ is an edge-preserving map, then $\theta$ is induced by a homeomorphism of $R$. We prove that if $R$ is closed and $\tau: \mathcal{ HT}(R) \rightarrow\mathcal{ HT}(R)$ is a rectangle preserving map, then $\tau$ is induced by a homeomorphism of $R$. We also prove that these homeomorphisms are unique up to isotopy when $(g, n) \neq (2, 0)$.
Categories: math.GT math.GR